This happens to be something I've accumulated a great deal of literature on over the past 20 years (several hundred papers). The literature is rather scattered and frequently in little known journals, and so various results are often rediscovered. Given what you've written and the time I have available right now, I'll limit myself to the following references. If you have more specific questions, I can probably point you to something in the literature, although it may take a day or two because all the papers and other materials I have are at home.
Charles Leonard Belna, Cluster sets of arbitrary real functions: A partial survey, Real Analysis Exchange 1 (1976), 7–20.
Ludek Zajicek, On cluster sets of arbitrary functions, Fundamenta Mathematica 83 (1973), 197-217.
http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83119.pdf
Andrew M. Bruckner I and Brian S. Thomson, Real Variable Contributions of G. C. Young and W. H. Young, Expositiones Mathematicae 19 (2001), 337-358.
http://classicalrealanalysis.info/documents/BT-YoungsArticle.pdf
Brian S. Thomson, Real Functions, Lecture Notes in Mathematics #1170, Springer-Verlag, 1985, viii + 229 pages. [See the chapter on cluster sets.]
Jaroslav Lukes, Jan Maly, and Ludek Zajicek, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics #1189, Springer-Verlag, 1986, x + 472 pages. [Various cluster set notions appear throughout.]
Cotinuous [sic] Functions and Limits [3-6 April 2007 sci.math thread.]
http://groups.google.com/group/sci.math/browse_thread/thread/50c64fb660caf79e
http://mathforum.org/kb/message.jspa?messageID=5626512
Basically an accumulation point has lots of the points in the series near it. A limit point has all (after some finite number) of points near it.
Think of the series $(-1+\frac 1{n^3})^n$. Both $-1$ and $1$ are accumulation points as there are entries very far out close to each. Neither is a limit because there are points very far out that are far away.
Best Answer
Terminology differs here, which might confuse you if see other answers online. But I'll follow Pugh here. We seem to be working in a metric space here, and using the notation $M_r(p)$ for an open ball of radius $r>0$ around $p$.
So whenever $S \subseteq X$, where $(X,d)$ is the metric space, then $p \in X$ is called a limit point of $S$ when for all $r>0$, $S \cap M_r(p) \neq \emptyset$; $p$ is called a cluster point of $S$ when for all $r > 0$ the set $S \cap M_r(p)$ is infinite, and $S$ condenses at $p$ (people also say that $p$ is a condensation point of $S$, which is more analogous to the previous names) whenever for all $r>0$ the set $S \cap M_r(p)$ is uncountable.
Note that if $p \in S$, then $M_r(p) \cap S$ will allways contain $p$ at least. So using this definition we see that all points of $S$ are themselves limit points of $S$, but there can be more, e.g. if $S = \{\frac{1}{n} \mid n \in \mathbb{N}^+\}$ (as a subset of the real numbers in the standard metric), then all points of $S$ are limit points of $S$, but $0 \notin S$ is too (and these are all the limit points of $S$). Now note that $1 \in S$ is not a cluster point of $S$, as $M_{\frac{1}{2}}(1)$ only intersects $S$ in $\{1\}$ and nowhere else. The same can be said for the other points $p$ of $S$: there exists some $r>0$ such that $S \cap M_r(p) = \{p\}$. Such a point is called an isolated point of $S$.
Now, in a metric space we have the following facts:
(1) if $p \in S$ is not an isolated point of $S$, then it is a cluster point of $S$.
Proof: suppose that $p \in S$ is not a cluster point of $S$. Then there is some $r>0$ such that $M_r(p) \cap S$ is finite, and so equals some set $\{p, p_1,\ldots,p_n\}$. Then $s = \min(r, d(p, p_1), \ldots, d(p, p_n)) > 0$ is well-defined (as a finite minimum of numbers > 0) and $M_{s}(p) \cap S = \{p\}$, as is easy to see. So $p$ is an isolated point of $S$. So all points of $S$ (which are all limit points of $S$) neatly divide into isolated points and cluster points of $S$.
(2) if $p \notin S$ is a limit point of $S$, then $p$ is a also a cluster point of $S$.
Proof: this is essentially the same argument. Suppose $p$ were not a cluster point, then for some $r>0$ we have $S \cap M_r(p) = \{p_1,\ldots,p_n\}$, where all $p_i \neq p$ (the set is not empty, as $p$ is a limit point by assumption). Defining $s = \min(d(p_1,p),\ldots,d(p_n,p))$, we see that again $s > 0$ and $M_s(p) \cap S = \emptyset$, contradiction.
So all limit points $p$ of $S$ are cluster points, except when $p \in S$ and $p$ is an isolated point of $S$. Another example where this occurs is for finite sets $S$ (no cluster points, and all points of the finite set are limit points and isolated points) and a set like $S = [0,1] \cup \{2\}$, where $2$ is a limit point, but isolated, so not a cluster point, and where all other points of $S$ are even condensation points of $S$.